Limit at negative infinity of a function with a radical $$\lim_{x \to -∞}{\sqrt{x^2 -x} + x}.$$ First I rationalized, to get $$\frac{x^2-x-x^2}{\sqrt{x^2-x}-x}$$ Then I wanted to factor the whole thing by the largest power(x) in the denominator, to get: $$\frac{-x/x}{1/x(\sqrt{x^2-x}-x)}$$ After simplifying: $$\frac{-1}{\sqrt{1-1/x}-1}$$ And evaluating at negative infinity: $$\frac{-1}{1-0-1}$$ But this is incorrect as the limit would not exist. Where did I go wrong? Am I right in my procedure? Thanks

> Where did I go wrong?

The following is **wrong** :

$$\frac{\sqrt{x^2-x}}{x}=\sqrt{1-\frac 1x}$$

Note that for $x\lt 0$ $$\frac{\sqrt{x^2-x}}{x}=\frac{\sqrt{x^2(1-\frac 1x)}}{x}=\frac{\sqrt{x^2}\sqrt{1-\frac 1x}}{x}=\frac{|x|\sqrt{1-\frac 1x}}{x}=\frac{\color{red}{-}x\sqrt{1-\frac 1x}}{x}=\color{red}{-}\sqrt{1-\frac 1x}$$